The proof of Theorem REF  uses the approach of geometric set theory [1]}. The model is constructed as a generic extension of the classical choiceless Solovay model [2]} by a rather canonical coloring poset (Definition REF ). The method uses an inaccessible cardinal to support the general framework. Removing the inaccessible cardinal in the spirit of [3]} requires plenty of improvisation, but in the given case is probably possible. Notation of the paper uses the set theoretic standard of [2]}, and in matters of geometric set theory, [1]}. DC is the Axiom of Dependent Choices. All theorems and definitions take place in ZFC set theory.
